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Multiple Phase Transitions in Long‐Range First‐Passage Percolation on Square Lattices
Author(s) -
Chatterjee Shirshendu,
S. Dey Partha
Publication year - 2016
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21571
Subject(s) - mathematics , square lattice , exponential growth , percolation (cognitive psychology) , combinatorics , square (algebra) , growth rate , directed percolation , exponential function , lattice (music) , percolation threshold , k nearest neighbors algorithm , norm (philosophy) , growth model , mathematical analysis , statistical physics , geometry , physics , critical exponent , quantum mechanics , scaling , mathematical economics , neuroscience , artificial intelligence , computer science , acoustics , law , political science , ising model , biology , electrical resistivity and conductivity
We consider a model of long‐range first‐passage percolation on the d ‐dimensional square lattice ℤ d in which any two distinct vertices x,y ∊ ℤ d are connected by an edge having exponentially distributed passage time with mean ‖ x – y ‖ α+ o (1) , where α > 0 is a fixed parameter and ‖·‖ is the l 1 –norm on ℤ d . We analyze the asymptotic growth rate of the set ß t , which consists of all x ∊ ℤ d such that the first‐passage time between the origin 0 and x is at most t as t → ∞. We show that depending on the values of α there are four growth regimes: (i) instantaneous growth for α < d , (ii) stretched exponential growth for α ∊ d ,2 d ), (iii) superlinear growth for α ∊ (2 d ,2 d + 1), and finally (iv) linear growth for α > 2 d + 1 like the nearest‐neighbor first‐passage percolation model corresponding to α =∞. © 2015 Wiley Periodicals, Inc.